Let the function $f$ defined on $\mathbb{R}$ by
$$f(x) = \ln\left(1 + \mathrm{e}^{-x}\right) + \frac{1}{4}x.$$
We denote $\mathscr{C}_f$ the representative curve of the function $f$ in an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath})$ of the plane.

\textbf{Part A}
\begin{enumerate}
  \item Determine the limit of $f$ at $+\infty$.
  \item We admit that the function $f$ is differentiable on $\mathbb{R}$ and we denote $f'$ its derivative function.\\
a. Show that, for all real $x$, $f'(x) = \dfrac{\mathrm{e}^x - 3}{4\left(\mathrm{e}^x + 1\right)}$.\\
b. Deduce the variations of the function $f$ on $\mathbb{R}$.\\
c. Show that the equation $f(x) = 1$ admits a unique solution $\alpha$ in the interval $[2;5]$.
\end{enumerate}

\textbf{Part B}

We will admit that the function $f'$ is differentiable on $\mathbb{R}$ and for all real $x$,
$$f''(x) = \frac{\mathrm{e}^x}{\left(\mathrm{e}^x + 1\right)^2}.$$
We denote $\Delta$ the tangent line to the curve $\mathscr{C}_f$ at the point with abscissa 0.\\
In the graph below, we have represented the curve $\mathscr{C}_f$, the tangent line $\Delta$, and the quadrilateral MNPQ such that M and N are the two points of the curve $\mathscr{C}_f$ with abscissas $\alpha$ and $-\alpha$ respectively, and Q and P are the two points of the line $\Delta$ with abscissas $\alpha$ and $-\alpha$ respectively.

\begin{enumerate}
  \item a. Justify the sign of $f''(x)$ for $x \in \mathbb{R}$.\\
b. Deduce that the portion of the curve $\mathscr{C}_f$ on the interval $[-\alpha; \alpha]$ is inscribed in the quadrilateral MNPQ.
  \item a. Show that $f(-\alpha) = \ln\left(\mathrm{e}^{-\alpha} + 1\right) + \dfrac{3}{4}\alpha$.\\
b. Prove that the quadrilateral MNPQ is a parallelogram.
\end{enumerate}